This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3.
3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar coordinates, and let $h : D \to D$ be the homeomorphism defined by $h(0) = 0,\:\: h(r, \theta) = (r, \theta + 2\pi r)$.
Find a homotopy $F$ from $h$ to the identity map such that the functions
$F|_{D \times \{t\}} \: D \times \{t\} \to D$
$0 \leq t \leq 1$, are all homeomorphisms.
4: With the terminology of Problem $3$, show that $h$ is homotopic to the identity map relative to $C$.
Define $F : D \times I \to D$
$F((r, \theta), t) = (r, \theta + 2\pi r(1 − t))$
$F((r, \theta), 0) = h(r, \theta)$
$F((r, \theta), 1) = i(r, \theta)$, where $i$ is the identity map.
$F$ is continuous and a homotopy between $h$ and $i$.
$F|_{D \times \{t\}} \: D \times \{t\} \to D$ are homeomorphisms since they are injective continuous maps from a compact space to a Hausdorff space.
Unfortunately i am not getting any further.